Translatieequivariant

Translatieequivariant, often called translation equivariant in English, describes a property of a function or operator with respect to translations. A system is translatieequivariant if translating its input by a certain amount results in the same translation being applied to the output. In practical terms, if T_a denotes a translation by a vector a, a function F is translation-equivariant when F(T_a(x)) equals T_a(F(x)) for all inputs x and all translations a. This means the operator consistently commutes with shifts.

Formally, translation equivariance can be described using group actions. The set of translations forms a group, and the action of this group on the input space induces a corresponding action on the output space. A map F is equivariant with respect to translations if F(g·x) = g·F(x) for every group element g (a translation) and input x. In discrete settings, translations shift grid indices; in continuous settings, they shift spatial coordinates. Equivariance is a structural property, separate from invariance, as the output changes in a predictable way under input shifts.

Convolution is a canonical example of a translation-equivariant operation. Convolving an input with a fixed kernel commutes with translations, which underpins the success of convolutional neural networks (CNNs). In neural network design, translational equivariance is often desired, but pooling and certain architectural choices can reduce exact equivariance. Extensions exist in the form of group-equivariant networks, which generalize equivariance to other symmetry groups beyond translations.

Applications of translatieequivariant principles appear in image processing, computer vision, physical simulations, and any domain where consistent behavior under shifts is important. The concept helps ensure that models respond to spatial patterns in a predictable and scalable way.